10 125
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 125's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
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10_125 is also known as the pretzel knot P(5,-3,2). |
Knot presentations
| Planar diagram presentation | X1425 X3849 X5,14,6,15 X20,16,1,15 X16,10,17,9 X18,12,19,11 X10,18,11,17 X12,20,13,19 X13,6,14,7 X7283 |
| Gauss code | -1, 10, -2, 1, -3, 9, -10, 2, 5, -7, 6, -8, -9, 3, 4, -5, 7, -6, 8, -4 |
| Dowker-Thistlethwaite code | 4 8 14 2 -16 -18 6 -20 -10 -12 |
| Conway Notation | [5,21,2-] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||
Length is 10, width is 3, Braid index is 3 |
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 [{12, 2}, {1, 10}, {8, 11}, {10, 12}, {9, 3}, {2, 8}, {7, 1}, {6, 9}, {5, 7}, {4, 6}, {3, 5}, {11, 4}] |
[edit Notes on presentations of 10 125]
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 125"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1425 X3849 X5,14,6,15 X20,16,1,15 X16,10,17,9 X18,12,19,11 X10,18,11,17 X12,20,13,19 X13,6,14,7 X7283 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 10, -2, 1, -3, 9, -10, 2, 5, -7, 6, -8, -9, 3, 4, -5, 7, -6, 8, -4 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 8 14 2 -16 -18 6 -20 -10 -12 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
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ConwayNotation[K]
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Out[8]=
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[5,21,2-] |
In[9]:=
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br = BR[K]
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KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
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Out[9]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(3,\{1,1,1,1,1,-2,-1,-1,-1,-2\})} |
In[10]:=
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{First[br], Crossings[br], BraidIndex[K]}
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KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
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KnotTheory::loading: Loading precomputed data in IndianaData`.
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Out[10]=
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{ 3, 10, 3 } |
In[11]:=
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Show[BraidPlot[br]]
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Out[11]=
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-Graphics- |
In[12]:=
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Show[DrawMorseLink[K]]
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KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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Out[12]=
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-Graphics- |
In[13]:=
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ap = ArcPresentation[K]
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Out[13]=
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ArcPresentation[{12, 2}, {1, 10}, {8, 11}, {10, 12}, {9, 3}, {2, 8}, {7, 1}, {6, 9}, {5, 7}, {4, 6}, {3, 5}, {11, 4}] |
In[14]:=
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Draw[ap]
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Out[14]=
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-Graphics- |
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
| Alexander polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3-2 t^2+2 t-1+2 t^{-1} -2 t^{-2} + t^{-3} } |
| Conway polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6+4 z^4+3 z^2+1} |
| 2nd Alexander ideal (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
| Determinant and Signature | { 11, 2 } |
| Jones polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^4+q^3-q^2+2 q-1+2 q^{-1} - q^{-2} + q^{-3} - q^{-4} } |
| HOMFLY-PT polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6-a^2 z^4-z^4 a^{-2} +6 z^4-4 a^2 z^2-4 z^2 a^{-2} +11 z^2-3 a^2-3 a^{-2} +7} |
| Kauffman polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^2 z^8+z^8+a^3 z^7+2 a z^7+z^7 a^{-1} -6 a^2 z^6-6 z^6-6 a^3 z^5-11 a z^5-5 z^5 a^{-1} +11 a^2 z^4+2 z^4 a^{-2} +13 z^4+10 a^3 z^3+17 a z^3+8 z^3 a^{-1} +z^3 a^{-3} -8 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} -15 z^2-4 a^3 z-8 a z-6 z a^{-1} -z a^{-3} +z a^{-5} +3 a^2+3 a^{-2} +7} |
| The A2 invariant | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{12}-q^{10}-q^8+q^4+2 q^2+3+2 q^{-2} + q^{-4} - q^{-8} - q^{-10} - q^{-12} } |
| The G2 invariant | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{60}+q^{56}-q^{54}-q^{48}-2 q^{44}-q^{40}-2 q^{38}-q^{36}-q^{34}-2 q^{32}-2 q^{28}-q^{26}+q^{24}-q^{22}+q^{20}+q^{16}+2 q^{14}+q^{12}+2 q^{10}+2 q^8+2 q^6+2 q^4+3 q^2+1+2 q^{-2} +2 q^{-4} + q^{-6} +2 q^{-8} +2 q^{-10} + q^{-14} +2 q^{-16} - q^{-18} + q^{-20} - q^{-24} + q^{-26} - q^{-28} - q^{-30} - q^{-34} - q^{-36} - q^{-38} -2 q^{-40} - q^{-44} - q^{-46} - q^{-50} - q^{-56} + q^{-72} } |
Further Quantum Invariants
Further quantum knot invariants for 10_125.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^9+q^3+q+ q^{-1} + q^{-3} - q^{-9} } |
| 2 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}-q^{24}-q^{18}-q^{16}+q^6+q^4+q^2+2+ q^{-2} + q^{-6} - q^{-12} - q^{-18} - q^{-20} + q^{-24} } |
| 3 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{57}+q^{53}+q^{51}-q^{47}+q^{43}+q^{41}-q^{37}-q^{35}-q^{27}-q^{25}-q^{23}-q^{17}-q^{15}+q^{13}+2 q^{11}+2 q^9+q^5+2 q^3+q+ q^{-3} + q^{-5} + q^{-13} - q^{-17} -2 q^{-19} - q^{-21} + q^{-23} +2 q^{-25} - q^{-27} -2 q^{-29} - q^{-31} +2 q^{-33} - q^{-37} + q^{-41} + q^{-43} - q^{-45} } |
| 5 | |
| 6 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{204}-q^{200}-q^{198}-q^{196}+2 q^{190}+2 q^{188}+q^{186}-q^{182}-2 q^{180}-3 q^{178}-q^{176}+2 q^{172}+3 q^{170}+3 q^{168}+2 q^{166}-q^{164}-2 q^{162}-3 q^{160}-3 q^{158}-2 q^{156}+2 q^{152}+3 q^{150}+3 q^{148}+2 q^{146}-2 q^{142}-3 q^{140}-3 q^{138}-2 q^{136}-q^{134}+q^{132}+2 q^{130}+4 q^{128}+3 q^{126}+q^{124}-q^{122}-4 q^{120}-5 q^{118}-5 q^{116}-q^{114}+3 q^{112}+6 q^{110}+7 q^{108}+6 q^{106}+q^{104}-5 q^{102}-7 q^{100}-7 q^{98}-2 q^{96}+4 q^{94}+9 q^{92}+9 q^{90}+5 q^{88}-q^{86}-8 q^{84}-10 q^{82}-7 q^{80}-q^{78}+4 q^{76}+8 q^{74}+8 q^{72}+2 q^{70}-4 q^{68}-8 q^{66}-9 q^{64}-7 q^{62}+6 q^{58}+8 q^{56}+5 q^{54}-6 q^{50}-10 q^{48}-7 q^{46}-q^{44}+7 q^{42}+10 q^{40}+9 q^{38}+2 q^{36}-6 q^{34}-9 q^{32}-6 q^{30}+q^{28}+7 q^{26}+11 q^{24}+6 q^{22}-2 q^{20}-7 q^{18}-8 q^{16}-2 q^{14}+4 q^{12}+9 q^{10}+6 q^8-q^6-5 q^4-6 q^2-1+4 q^{-2} +7 q^{-4} +5 q^{-6} - q^{-8} -4 q^{-10} -5 q^{-12} -2 q^{-14} + q^{-16} +4 q^{-18} +2 q^{-20} - q^{-22} -2 q^{-24} - q^{-26} +2 q^{-28} + q^{-30} -2 q^{-34} -3 q^{-36} - q^{-38} + q^{-40} +5 q^{-42} +5 q^{-44} +3 q^{-46} - q^{-48} -4 q^{-50} -5 q^{-52} -6 q^{-54} -3 q^{-56} +2 q^{-58} +7 q^{-60} +9 q^{-62} +8 q^{-64} + q^{-66} -7 q^{-68} -14 q^{-70} -13 q^{-72} -6 q^{-74} +5 q^{-76} +16 q^{-78} +15 q^{-80} +8 q^{-82} -5 q^{-84} -15 q^{-86} -18 q^{-88} -9 q^{-90} +6 q^{-92} +14 q^{-94} +14 q^{-96} +5 q^{-98} -4 q^{-100} -12 q^{-102} -9 q^{-104} +6 q^{-108} +8 q^{-110} +5 q^{-112} + q^{-114} -4 q^{-116} -5 q^{-118} -2 q^{-120} + q^{-122} +2 q^{-124} +2 q^{-126} + q^{-128} -2 q^{-130} -4 q^{-132} -2 q^{-134} + q^{-138} +2 q^{-140} +2 q^{-142} + q^{-144} - q^{-146} - q^{-148} } |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{12}-q^{10}-q^8+q^4+2 q^2+3+2 q^{-2} + q^{-4} - q^{-8} - q^{-10} - q^{-12} } |
| 1,1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{36}+2 q^{32}-2 q^{30}+2 q^{28}-2 q^{26}+2 q^{24}-2 q^{22}-4 q^{20}-2 q^{18}-6 q^{16}-5 q^{12}+2 q^{10}+6 q^6+5 q^4+8 q^2+8+6 q^{-2} +5 q^{-4} -2 q^{-6} -2 q^{-8} -4 q^{-10} -5 q^{-12} -2 q^{-14} -2 q^{-16} +2 q^{-20} -2 q^{-26} + q^{-36} } |
| 2,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{34}+q^{32}+q^{30}-q^{24}-2 q^{22}-3 q^{20}-3 q^{18}-2 q^{16}-2 q^{14}-2 q^{12}-q^{10}+q^8+3 q^6+5 q^4+6 q^2+8+5 q^{-2} +4 q^{-4} - q^{-8} -3 q^{-10} -2 q^{-12} -3 q^{-14} -3 q^{-16} -2 q^{-18} - q^{-20} - q^{-24} + q^{-26} + q^{-28} + q^{-30} } |
A3 Invariants.
| Weight | Invariant |
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| 0,1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{26}+q^{22}-q^{18}-2 q^{16}-3 q^{14}-4 q^{12}-5 q^{10}-2 q^8+5 q^4+7 q^2+10+8 q^{-2} +5 q^{-4} + q^{-6} -2 q^{-8} -4 q^{-10} -4 q^{-12} -3 q^{-14} -2 q^{-16} - q^{-18} + q^{-30} } |
| 1,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{15}-q^{13}-2 q^{11}-q^9-q^7+q^5+3 q^3+4 q+4 q^{-1} +3 q^{-3} + q^{-5} - q^{-7} - q^{-9} -2 q^{-11} - q^{-13} - q^{-15} } |
| 1,0,1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{44}+2 q^{40}+q^{36}+q^{32}+q^{30}+q^{26}-4 q^{24}-2 q^{22}-9 q^{20}-7 q^{18}-14 q^{16}-11 q^{14}-11 q^{12}-5 q^{10}+5 q^8+11 q^6+24 q^4+22 q^2+30+18 q^{-2} +16 q^{-4} +3 q^{-6} -6 q^{-8} -11 q^{-10} -17 q^{-12} -13 q^{-14} -15 q^{-16} -5 q^{-18} -4 q^{-20} +4 q^{-22} +3 q^{-24} +5 q^{-26} +3 q^{-28} - q^{-30} -2 q^{-34} + q^{-48} } |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{32}+q^{30}+2 q^{28}+2 q^{26}+2 q^{24}-3 q^{20}-6 q^{18}-9 q^{16}-11 q^{14}-12 q^{12}-8 q^{10}-3 q^8+5 q^6+11 q^4+18 q^2+21+19 q^{-2} +14 q^{-4} +8 q^{-6} -6 q^{-10} -9 q^{-12} -10 q^{-14} -10 q^{-16} -7 q^{-18} -4 q^{-20} -2 q^{-22} - q^{-24} + q^{-26} +2 q^{-28} + q^{-30} + q^{-32} + q^{-34} + q^{-36} } |
| 1,0,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{18}-q^{16}-2 q^{14}-2 q^{12}-2 q^{10}-q^8+q^6+3 q^4+5 q^2+5+5 q^{-2} +3 q^{-4} + q^{-6} - q^{-8} -2 q^{-10} -2 q^{-12} -2 q^{-14} - q^{-16} - q^{-18} } |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{26}-q^{22}-q^{18}-q^{14}+q^{10}+2 q^6+q^4+3 q^2+2 q^{-2} + q^{-4} + q^{-6} - q^{-14} - q^{-18} - q^{-30} } |
| 1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{44}+q^{36}-q^{32}-q^{30}-q^{26}-2 q^{24}-2 q^{22}-q^{20}-q^{18}-2 q^{16}-q^{14}-q^{12}+q^{10}+q^8+3 q^6+2 q^4+4 q^2+4+4 q^{-2} +2 q^{-4} +3 q^{-6} +2 q^{-8} + q^{-10} - q^{-12} - q^{-14} - q^{-16} - q^{-18} -2 q^{-20} -2 q^{-22} - q^{-24} - q^{-26} - q^{-28} - q^{-30} + q^{-48} } |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{34}+q^{30}+q^{26}-q^{24}-q^{22}-3 q^{20}-4 q^{18}-5 q^{16}-6 q^{14}-4 q^{12}-3 q^{10}+q^8+3 q^6+8 q^4+9 q^2+12+9 q^{-2} +8 q^{-4} +4 q^{-6} + q^{-8} -2 q^{-10} -4 q^{-12} -5 q^{-14} -5 q^{-16} -3 q^{-18} -3 q^{-20} - q^{-22} - q^{-24} + q^{-42} } |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{60}+q^{56}-q^{54}-q^{48}-2 q^{44}-q^{40}-2 q^{38}-q^{36}-q^{34}-2 q^{32}-2 q^{28}-q^{26}+q^{24}-q^{22}+q^{20}+q^{16}+2 q^{14}+q^{12}+2 q^{10}+2 q^8+2 q^6+2 q^4+3 q^2+1+2 q^{-2} +2 q^{-4} + q^{-6} +2 q^{-8} +2 q^{-10} + q^{-14} +2 q^{-16} - q^{-18} + q^{-20} - q^{-24} + q^{-26} - q^{-28} - q^{-30} - q^{-34} - q^{-36} - q^{-38} -2 q^{-40} - q^{-44} - q^{-46} - q^{-50} - q^{-56} + q^{-72} } |
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Computer Talk
The above data is available with the Mathematica package
KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["10 125"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3-2 t^2+2 t-1+2 t^{-1} -2 t^{-2} + t^{-3} } |
In[5]:=
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Conway[K][z]
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Out[5]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6+4 z^4+3 z^2+1} |
In[6]:=
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Alexander[K, 2][t]
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KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
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Out[6]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 11, 2 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^4+q^3-q^2+2 q-1+2 q^{-1} - q^{-2} + q^{-3} - q^{-4} } |
In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6-a^2 z^4-z^4 a^{-2} +6 z^4-4 a^2 z^2-4 z^2 a^{-2} +11 z^2-3 a^2-3 a^{-2} +7} |
In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^2 z^8+z^8+a^3 z^7+2 a z^7+z^7 a^{-1} -6 a^2 z^6-6 z^6-6 a^3 z^5-11 a z^5-5 z^5 a^{-1} +11 a^2 z^4+2 z^4 a^{-2} +13 z^4+10 a^3 z^3+17 a z^3+8 z^3 a^{-1} +z^3 a^{-3} -8 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} -15 z^2-4 a^3 z-8 a z-6 z a^{-1} -z a^{-3} +z a^{-5} +3 a^2+3 a^{-2} +7} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 125"];
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In[4]:=
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{A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[4]=
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{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3-2 t^2+2 t-1+2 t^{-1} -2 t^{-2} + t^{-3} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^4+q^3-q^2+2 q-1+2 q^{-1} - q^{-2} + q^{-3} - q^{-4} } } |
In[5]:=
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DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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Out[5]=
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{} |
In[6]:=
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DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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Out[6]=
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{} |
Vassiliev invariants
| V2 and V3: | (3, 0) |
| V2,1 through V6,9: |
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V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.
Khovanov Homology
| The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 2 is the signature of 10 125. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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| Integral Khovanov Homology
(db, data source) |
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The Coloured Jones Polynomials
The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1}
-dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)}
)
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n} |
| 2 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{11}-q^{10}-q^9+q^8-q^6+q^4-q^3+q^2+2 q^{-1} - q^{-2} +2 q^{-4} -2 q^{-5} +2 q^{-7} -2 q^{-8} - q^{-9} +2 q^{-10} - q^{-11} - q^{-12} + q^{-13} } |
| 3 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{21}+2 q^{20}-q^{18}-2 q^{17}+3 q^{16}+2 q^{15}-4 q^{14}-3 q^{13}+4 q^{12}+5 q^{11}-5 q^{10}-5 q^9+3 q^8+6 q^7-4 q^6-4 q^5+2 q^4+6 q^3-4 q^2-3 q+2+5 q^{-1} -3 q^{-2} -2 q^{-3} + q^{-4} +4 q^{-5} - q^{-6} -2 q^{-7} +2 q^{-9} - q^{-10} - q^{-11} + q^{-13} - q^{-14} - q^{-15} + q^{-16} + q^{-17} - q^{-18} -2 q^{-19} + q^{-20} +2 q^{-21} -2 q^{-23} + q^{-25} + q^{-26} - q^{-27} } |
| 4 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{33}+q^{32}+q^{31}-q^{29}-2 q^{28}+3 q^{27}+q^{26}-q^{25}-4 q^{24}-q^{23}+7 q^{22}+2 q^{21}-3 q^{20}-8 q^{19}-q^{18}+10 q^{17}+4 q^{16}-2 q^{15}-11 q^{14}-3 q^{13}+11 q^{12}+4 q^{11}-q^{10}-10 q^9-3 q^8+9 q^7+3 q^6-8 q^4-3 q^3+8 q^2+3 q+1-7 q^{-1} -4 q^{-2} +6 q^{-3} +4 q^{-4} +2 q^{-5} -5 q^{-6} -5 q^{-7} +4 q^{-8} +3 q^{-9} +3 q^{-10} -2 q^{-11} -5 q^{-12} + q^{-13} + q^{-14} +3 q^{-15} + q^{-16} -4 q^{-17} - q^{-18} - q^{-19} + q^{-20} +3 q^{-21} -2 q^{-22} - q^{-24} - q^{-25} +3 q^{-26} -2 q^{-27} + q^{-28} - q^{-30} +3 q^{-31} -3 q^{-32} +4 q^{-36} -2 q^{-37} - q^{-38} - q^{-39} - q^{-40} +3 q^{-41} - q^{-44} - q^{-45} + q^{-46} } |
| 5 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{51}-q^{50}-q^{48}-q^{47}+q^{46}+2 q^{45}+q^{44}-q^{43}-q^{42}-2 q^{41}+q^{40}+q^{39}+q^{38}+q^{37}+q^{36}-q^{35}-2 q^{34}-5 q^{33}-q^{32}+3 q^{31}+8 q^{30}+5 q^{29}-4 q^{28}-10 q^{27}-7 q^{26}+q^{25}+10 q^{24}+11 q^{23}-10 q^{21}-10 q^{20}-2 q^{19}+8 q^{18}+11 q^{17}+2 q^{16}-8 q^{15}-9 q^{14}-q^{13}+6 q^{12}+9 q^{11}-7 q^9-7 q^8+5 q^6+8 q^5-4 q^3-6 q^2-2 q+2+7 q^{-1} +3 q^{-2} - q^{-3} -5 q^{-4} -4 q^{-5} -2 q^{-6} +6 q^{-7} +5 q^{-8} +3 q^{-9} -3 q^{-10} -6 q^{-11} -5 q^{-12} +3 q^{-13} +6 q^{-14} +6 q^{-15} -6 q^{-17} -7 q^{-18} - q^{-19} +4 q^{-20} +6 q^{-21} +4 q^{-22} -3 q^{-23} -6 q^{-24} -3 q^{-25} - q^{-26} +3 q^{-27} +5 q^{-28} - q^{-30} -2 q^{-31} -3 q^{-32} - q^{-33} +2 q^{-34} + q^{-35} +2 q^{-36} + q^{-37} - q^{-38} - q^{-39} - q^{-40} - q^{-41} + q^{-42} + q^{-43} + q^{-44} - q^{-47} - q^{-51} + q^{-53} + q^{-54} + q^{-55} -2 q^{-57} -2 q^{-58} + q^{-60} + q^{-61} +2 q^{-62} -2 q^{-64} - q^{-65} + q^{-68} + q^{-69} - q^{-70} } |
| 6 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{71}+2 q^{69}+q^{68}-q^{66}-q^{65}-3 q^{64}-2 q^{63}+4 q^{62}+4 q^{61}+q^{60}-q^{59}-2 q^{58}-6 q^{57}-5 q^{56}+5 q^{55}+9 q^{54}+5 q^{53}+2 q^{52}-4 q^{51}-12 q^{50}-14 q^{49}+2 q^{48}+17 q^{47}+14 q^{46}+11 q^{45}-4 q^{44}-20 q^{43}-29 q^{42}-7 q^{41}+20 q^{40}+24 q^{39}+24 q^{38}+3 q^{37}-19 q^{36}-40 q^{35}-18 q^{34}+13 q^{33}+23 q^{32}+31 q^{31}+11 q^{30}-12 q^{29}-39 q^{28}-20 q^{27}+8 q^{26}+18 q^{25}+28 q^{24}+12 q^{23}-11 q^{22}-36 q^{21}-16 q^{20}+10 q^{19}+18 q^{18}+24 q^{17}+10 q^{16}-13 q^{15}-35 q^{14}-14 q^{13}+11 q^{12}+19 q^{11}+21 q^{10}+9 q^9-12 q^8-32 q^7-12 q^6+8 q^5+16 q^4+18 q^3+10 q^2-9 q-26-10 q^{-1} +5 q^{-2} +11 q^{-3} +13 q^{-4} +11 q^{-5} -5 q^{-6} -19 q^{-7} -7 q^{-8} +5 q^{-10} +7 q^{-11} +12 q^{-12} -11 q^{-14} -2 q^{-15} -4 q^{-16} - q^{-17} +9 q^{-19} +3 q^{-20} -3 q^{-21} +5 q^{-22} -3 q^{-23} -4 q^{-24} -8 q^{-25} +3 q^{-26} + q^{-28} +11 q^{-29} +2 q^{-30} - q^{-31} -10 q^{-32} -3 q^{-33} -7 q^{-34} - q^{-35} +12 q^{-36} +6 q^{-37} +5 q^{-38} -4 q^{-39} -3 q^{-40} -11 q^{-41} -6 q^{-42} +6 q^{-43} +3 q^{-44} +7 q^{-45} +3 q^{-46} +3 q^{-47} -7 q^{-48} -6 q^{-49} + q^{-50} -3 q^{-51} +2 q^{-52} +3 q^{-53} +5 q^{-54} - q^{-55} - q^{-56} +2 q^{-57} -4 q^{-58} - q^{-59} - q^{-60} + q^{-61} - q^{-62} +5 q^{-64} -2 q^{-65} + q^{-66} - q^{-68} -2 q^{-69} -2 q^{-70} +4 q^{-71} -3 q^{-72} + q^{-73} + q^{-74} + q^{-75} - q^{-77} +4 q^{-78} -4 q^{-79} - q^{-80} - q^{-81} +5 q^{-85} - q^{-86} - q^{-88} - q^{-89} -2 q^{-90} - q^{-91} +3 q^{-92} + q^{-94} - q^{-97} - q^{-98} + q^{-99} } |
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.
Modifying This Page
| Read me first: Modifying Knot Pages
See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top. |
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